Implementation Notes on Diffusivity with Steering Level Suppression (Gokhan and John Marshall)
12 November 2014
Starting with \[K=u_{rms}∗L_{mix}\]
the general form for diffusivity, \(K\), is given by equation (6) of Bates et al. (2014). Namely,
\[L_{mix} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} \]
Here,
\(u_{rms}\) is the root-mean-square (rms) eddy velocity;
\(L_{mix}\) is the mixing length;
\(L_{eddy}\) is the eddy diameter (depth independent);
\(u_{mean}\) is the mean zonal velocity (resolved);
\(c\) is the zonal eddy phase speed (depth independent);
- \(\Gamma = 0.35\)
\(b1 \sim 4\). ###\(\quad\quad\quad\quad\quad\quad\) Eddy Length Scales: Rossby deformation radius = \(L_r = {c_r \over |f|}\)
Equatorial Rossby deformation radius \(= L_{req} = {\sqrt{c_r \over 2\beta}}\)
Rhines scale \(= L_{Rh} = {\sqrt{u_{rms} \over \beta}} \sim {\sigma_{vi} \over \beta}\)
\(c_r\) is the first baroclinic wave speed computed following equation (2.2) of Chelton et al. (1998) with \(m=1\);
\(f\) is the Coriolis parameter;
\(\beta\) is the latitudinal variation of the Coriolis parameter; and
\(\sigma_{vi}\) is the Eady growth rate given by
\(\sigma_{vi} = {f \over \sqrt{R_i}}\)
with \(R_i\) the vertically integrated (over the 100 – 2000 m depth range) Richardson number.
So, any one these length scales could be used as an eddy length scale. An alternative is
\(L_{eddy} = min (L_r, L_{req}, L_{Rh}).\)
\(\quad\quad\quad\quad\quad\quad\) Eddy Velocity:
\({u_{rms} = alpha*\sigma_{vi}*L_r}\)
where \(\sigma_{vi}\) is the Eady growth rate based on local Richardson number and \(\alpha\) is a scaling constant.
\(\quad\quad\quad\quad\quad\quad\) Zonal phase speed:
\(c = - \beta * L_r^2\)
References:
Bates, M., R. Tulloch, J. Marshall, and R. Ferrari, 2014: Rationalizing the spatial distribution of mesoscale eddy diffusivity in terms of mixing length theory. J. Phys. Oceanogr., 44, 1523-1540, doi: 10.1175/JPO-D-13-0130.1.
Chelton, D. B., R. A. deSzoeke, M. G. Schlax, K. E. Naggar, and N. Siwertz, 1998: Geographical variability of the first baroclinic Rossby radius of deformation. J. Phys. Oceanogr., 28, 433-460.
Tullock, R., J. Marshall, and K. S. Smith, 2009: Interpretation of the propagation of surface altimetric observations in terms of planetary waves and geostropic turbulence. J. Geophys. Res., 114, C02005, doi: 10.1029/2008JC005055.
Questions:
In the 2D implementation, \(u_{rms}\) and \(u_{mean}\) specifications: upper-ocean vertically or integrated or at \(z = 0\)?
A: \(U_{rms}\) is not depth dependent; both \(u_{rms}\) and \(u_{mean}\) are for surface only
In the 2D implementation, vertical profile will be specified by \(N2(z)\)?
A: Yes, \(N^2(z) \over N_{ref}(z)\) to be more precise
Local \(R_i\) use imbedded in sigma in \(u_{rms}\) calculation?
A: No, \(u_{rms}\) is depth independent
alpha = ?
A: trial and error
Cancellation of \(f\)’s in \(u_{rms}\) calculation?
Zonal phase speed equation correct? Both \(\beta\) and \(L_r\) will be positive, >producing \(c < 0\) always. This appears to be in contrast with Tullock et al. (2009).
A: What is plotted in Bates is (U-c)
\(\color{green}{\quad CESM \quad Implementation \quad of \quad Steering \quad Parameterization}\)
During the development of the parameterization several tuning modifications were made to the terms listed above to better match the observations. * New implementation of Eady Growth rate (sigma) that avoids undefined values at the equator. See alternate derivation at end of this page. >\(\sigma_{vi} = {f \over \sqrt{R_i}}\) is replace by \(\sigma_{vi} = \quad {{\require\cancel f m^2} \over \cancel f N}\)
Don't use Rhine Scale in calculation of \(L_{eddy}\) >\(L_{eddy} = min (L_r, L_{req}, \cancel{L_{Rh}}).\)
Eddy phase speed calculation modified to avoid undefined value at the equator and limited to better match Bates paper.
>\(\cancel{c = - \beta * {L_r^2}}\) is replace with \(c = max(- \beta * L_{eddy}^2,-20)\)\(u_{rms}\) is hard to capture in this implememtation due to the coarse model resolution and use of mean vs resolved states. We imposed a \(5 cm/sec\) minimum on \(U_{rms}\) to tune to surface obs of Bates and included a scaling constant \(\alpha\) to handle the bias introduced by using mean state instead of resolved state >\(u_{rms} = max(5.,alpha*\sigma_{vi}*L_r)\)
\(\alpha=4\) The scaling constant for \(U_{rms}\)
- The Eddy diffusivity calculated by our implementation was weak so be bumped up \(\Gamma\) from .35 to 1.75 (5x increase) > \(\Gamma=.35\) replated with \(\Gamma = 1.75\)
\(\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} \)
NOTES: >\[K=u_{rms}∗{\Gamma * \color{red}{L_{eddy}} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} \] > >The Eddy Length scale is parameterized as \(L_{eddy} = min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}}).\) > >Since the two length scales we are using in the parameterization of \(L_{eddy}\) depend on the Baroclinic wave speed \(c_r\) we will first check \(c_r\) against Chelton 1998.
\(\quad\quad\quad\quad\quad\quad\quad\) First Baroclinic Wave Speed \(c_r\) CESM
\(\quad\quad\quad\quad\quad\quad\) Chelton Baroclinic gravity wave phase speed
The barclinic wave speed looks reasonable so lets compare the Paramaterized Rossby Radius to Chelton
\(\quad\quad\quad\quad\quad\quad\) First Baroclinic Rossby Radius CESM \(\quad L_{eddy}= min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}}).\)
\(\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})\)
NOTES:
\(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - \color{red}{c}|^2 /u_{rms}^2 (z=0)} \)
\(\require{cancel}\cancel{c = - \beta * {L_r^2}}\) \(L_r\) too high at equator
\(c = - \beta * L_{eddy}^2\)
\(\quad\quad\quad\quad\quad\) Eddy Phase speed CESM
Compared to Hughes, the phase speed is much too high near the equator.
The first tuning mod is to limit the phase speed to 20cm/s.
\(\quad \quad \quad \quad \)Eddy Phase speed CESM \((c = max(- \beta * L_{eddy}^2,-20) )\)
\(\quad\quad\quad\quad\quad\quad\) Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09
\(\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})\)
NOTES:
\(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |\color{red}{u_{mean}} - c|^2 /u_{rms}^2 (z=0))} \)
Here \(U_{mean}\) is just CESM surface velocity.
\(\quad\quad \quad \quad\quad \) Zonal Velocity CESM
\(\color{green}{(U-c)\quad Term}\)
NOTES: \(c = max(- \beta * L_r^2,-20)\) )
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) Bates et al
\(\color{green}{(U-c)^2 \quad Term}\)
NOTES: \(c = max(- \beta * L_{eddy}^2,-20)\)
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) Bates et al
\(\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma}\)
NOTES: >The original derivation of \(\sigma_{vi}\) is undefined at the equator because of \(f\)
\(\sigma_{vi} = {f \over \sqrt{R_i}}\)
The new derivation of \(\sigma_{vi}\):
$ R_i = {f^2 N^2 _}= $
\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)
\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) CESM
\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) Bates et al
\(\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}\)
NOTES: >\(K=\color{red}{u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)} \) > >\(u_{rms} = alpha*{\sigma_{vi}}*L_{eddy}\) > >alpha (scaling constant) = 4 > >\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\) > >\(u_{rms}\) is limited to 5 cm/s $ max(u_{rms},5.)$
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity \((u_{rms})\quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} \) Bates et al
\(\color{green}{u_{rms}^2 \quad Term} \quad\)
NOTES: >\(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /\color{red}{u_{rms}^2} (z=0)} \) > >\(u_{rms}^2\) is a surface value
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity Squared \((u_{rms}^2)\quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 \) Bates et al
\(\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}\)
NOTES: >\(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * \color{red}{|u_{mean} - c|^2 /{u_{rms}^2} (z=0)}} \) > > >\(c = max(- \beta * L_{eddy}^2,-20)\) > >\(u_{rms} = max(u_{rms},5.)\) > >\(u_{rms}^2\) is a surface value
\(\quad\quad\quad\quad\quad\quad\) \({(U-c)}^2 \over {u_{rms}^2} \quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} \) Bates et al
\(\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}\)
NOTES: > >\(c = max(- \beta * L_{eddy}^2,-20)\) > >\(u_{rms} = max(u_{rms},5.)\) > >\(u_{rms}^2\) is a surface value > >\(b1\) (scaling constant) = 4.
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor Bates
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) Bates
\(\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}\)
NOTES: >\(\color{red}{L_{mix}} = \Gamma * L_{eddy} * Suppression\) > >\(Suppression= {1 \over (1 + b1 * |u_{mean} - c|^2 /u_{rms (z=0)}^2 )}\) > >\(\color{red}{L_{mix}} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\) > >\(\Gamma = 1.75\) (Tuning mod: the original Gamma of .35 produced a Kappa with the correct structure but too weak.;
\(\quad\quad\quad\quad\quad\quad\quad\) LMIX CESM
\(\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})\)
NOTES: >###\(\color{red}{K}=u_{rms}∗L_{mix}\)
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) Bates
\(\quad\quad\quad\quad\quad\quad\quad\) (Zonal x Depth) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\) K limited to (100 < K < 10000)
Here is the Bate's version
\(\quad\quad\quad\quad\quad\quad\quad\) Zonal average of the N2 normalized scaling CESM
The alternate derivation of \(\sigma_{vi}\):
$ R_i = {N^2 }$
$ N^2 = {{-g _0 }} $
After hydrostatic and geostrophic approximations
\(f \frac {\partial v}{\partial z} = {{-g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad f \frac {\partial u}{\partial z} = {{g \over \rho_0 }\frac {\partial \rho}{\partial y}} \)
so
\(\frac {\partial v}{\partial z} = {{-1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad \frac {\partial u}{\partial z} = {{1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial y}} \)
\(\therefore\)
$ R_i = {f^2 N^2 _}= $
\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)
\(RX_1 = RX_{east} = \Delta\rho_x = \rho_{i+1,j} - \rho_{i,j}\)
\(RY_1 = RY_{north} = \Delta\rho_y = \rho_{i,j+1} - \rho_{i,j}\)
\(RZ_1 = RZ_{k+1} = \Delta\rho_z = \rho_{k} - \rho_{k+1}\)
\(\displaystyle{1 \over L_{R_i}} \displaystyle\int_{2000m}^{100m} \left\lbrace { {-g\over\rho_0}{\frac {\partial \rho} {\partial z}} \over { {g^2 \over \rho_0^2 } \left[( \frac{\partial \rho}{\partial y})^2 + ( \frac{\partial \rho}{\partial z})^2\right] } } \right\rbrace dz\)
Note: missing \(f^2\) which will be cancelled when forming \(\sigma_{vi}\)
\(\quad\quad\) so \(\cdots\) this is not \(R_i\)
Implementation notes
Numerator : Top \(= -grav * RZ_{SAVE}(\cdots k+1) * dzwr(k)\)
Denominator :
$$
$$
\(work3 = {\left( TOP \over (grav^2*(work1+work2))\right)}*dzw(k)\)
Notes:
1)Need to be careful at top and bottom of ocean
2)Accurate dzw(k) for each (i,j) to form $L_{R_i}$
3) When constructing $sigma$ itself, use $RZ_{SAVE}$ with a minimum N value
4) use eps2